An Improved Nonlinear Speed Controller for Series DC Motors

Transcription

1 Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 8 An Improved Nonlinear Speed Controller for Series DC Motors Dongbo Zhao Ning Zhang Department of Electrical Engineering and Applied Electronic Technology, Tsinghua University, Beijing, 84, P.R.China ( zdb4@mails.tsinghua.edu.cn) Department of Electrical Engineering and Applied Electronic Technology, Tsinghua University, Beijing, 84, P.R.China ( zhangn3@mails.tsinghua.edu.cn) Abstract: The issue of speed tracking control for series DC motors is addressed. Based on the classic backstepping design technique, an improved recursive nonlinear controller is proposed to improve the transient response. In the design, two additional class K functions as design functions are adopted to achieve desirable varying decay rate. Application of this strategy substantiallyimproves the transientresponse andconvergence rate withoutremarkably increasing the controller gains. Series DC motors with jumping load torques are also studied. The dynamics of such a motor with jumping load torques are modeled as a switched system. Aswitchingcontrollerbasedontheimprovednonlineardesignmethodis presented. Simulation results demonstrate the effectiveness of the proposed method. 1. INTRODUCTION The control of a series DC motor is of great importance because of extensive engineering applications of series DC motors. Meanwhile, since the dynamics of a series DC motor can be described by a typical nonlinear system with lower triangular structure, the study of series DC motors from the control point of view also provides many challenging theoretical issues to address. The traditional way to control a series DC motor is to adopt a linear controller which is based on the linear approximation of the original nonlinear model around a nominal operation point (Rashid (1993)). However, this linear controller is not applicable to the case when the motor operates on wide dynamical regimes. To overcome this drawback, multiple linear controllers are used to control the motor, where each linear controller is applied to a certain neighborhood of a operating point (Sira-Ramirez (199)). As a result, a piecewise linear controller is formed. Thiscontrolstrategy reallyworksin some cases and is easy to implement in engineering but may not produce satisfactory solutions in some other cases. Recent advances in nonlinear control systems makes it possible to apply nonlinear controllers. Exact feedback linearization technique was introduced to the controller design of series DC motors (Chiasson (1994); Olivier (1991)). This technique cancels allnonlinearitiesanransformsthenonlinearsystem into a linear controllable one by coordinates transformation and feedback, and thus standard design means for linear systems can be applied in the new coordinates (Krstic et al. (1995)). However, theexact feedbacklinearizationmethod requires precise models and often cancel some useful nonlinearity. Backstepping is another effective nonlinear design method which preservers certain useful nonlinearity and gives the controller systematically. Also, compared with exact feedback linearization method, backstepping often has better robustness. Therefore, nonlinear control based on backstepping has attracted a lot of attention. Manyresults usingbacksteppingtocontroldc motors are available in the literature, see, for example, Burridge et al. (3); Wang et al. (6). For speed tracking control of series DC motors, nonlinear control approaches, such as backstepping and feedback linearization methods, usually give asymptotic tracking results with constant decay rates. If we want a fast transient response, large control gains must be applied which in many cases is impossible or means high cost. How to have a good transient response without remarkably increasing the control gains is a significant problem. Since uncertainties and disturbances are unavoidable in practice, changing load torques must be taken into account when designing a controller for a series DC motor. A number of advanced robust control methods have been exploited to handle uncertainties and disturbances in motor systems. Forexample,a robusttrackingcontrolwas given for varying parameters in Liu (6); tracking periodic signals was addressed for uncertain parameters in Wai (1); An adaptive backstepping method was applies to the problem of motion control (Yu et al. (1)); A robust feedbacklinearizationapproachwas presentedin Bogosyan et al. (). On the other hand, loads can be changed abruptly, whichcauses jumpingloaorques. Mostrobust control methods which are effective to deal with slowly varying parameters are not applicable to control of jumping load torques. Moreover, transient responses are even more important when we control a motor with jumping load torques because there is often no sufficient time to have a transient response which is good enough before a new load torque is applied. In this paper we study the problem of speed tracking control for series DC motors. An improved recursive nonlinear /8/$. 8 IFAC /76-5-KR

2 17th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-11, 8 controller is proposed to improve the transient response. In the design, two additional class K functions as design functions are adopted to adjust the response and decay rates. Series DC motors withjumpingloaorques are also studied. The dynamicsofsuchamotorcanbe modeledas aswitchedsystem so that thedesign methodsfor switched systems are applicable. Inparticular, aswitchingcontroller based on the improved design method is presented. 2. MOTOR DYNAMICS Consider a series DC motor shown in Figure 1 below. Fig. 1. Series DC motor If we neglect magnetic saturation in field circuit, the motor can be modeled as (refer to Liu et al. (1999) for details). L di = u Ri Miω, J dω = Mi2 T L, (1) where the physical meaning of the quantities are as follows. i: armature current (or filed current) u: terminal control voltage ω: rotational speed of the motor L: total armature and field current inductance R: total armature and field circuit resistance J: moment of inertia associated with both motor and the load M: motor constant T L : load torque T e = Mi 2 and E = Miω. For notational convenience, we set x 1 = ω, x 2 = i. Then, the system (1) becomes dx 1 = M J x2 2 T L J, dx 2 = 1 L (Rx 2 + Mx 1 x 2 ) + 1 (2) L u. The control goal is to design a feedback controller u such that the speed of the motor x 1 tracks a constant desired speed ω r. 3. CONTROLLER DESIGN In this section, we develop an improved backstepping design method for the tracking control of the series DC motor. We first recall the classic backstepping design for the series DC motor. 3.1 Classic Backstepping Design The classic backsteppingdesign for (2) canbe summarized as follows (Liu et al. (1999)). Step 1. Let e = x 1 ω r. Its derivative along the system is ė = ẋ 1 = 1 J (Mx2 2 T L ). Define the Lyapunov function for this step by V 1 = 1 2 e2, whose derivative along the system is In order to have V 1 = 1 J e(mx2 2 T L ). V 1 = k 1 e 2 for some constant k 1 >, the virtual control x 2 2 should be Step 2. Set J M k 1e + T L M. η = M(x J M k 1e T L M ) = Mx Jk 1 e T L. Define the Lyapunov function for Step 2 as V 2 = V η2 = 1 2 e η2. Differentiating V 2 along the system yields V 2 = k 1 e J eη +η[ 2Mx 2 L (u (Rx 2 + Mx 1 x 2 )) +k 1 η k1je]. 2 Choosing u = Rx 2 + Mx 1 x 2 + L ( k 1 η + k 2Mx 1Je 2 1 ) 2 J e k 2η with a constant k 2 >, we have V 2 k 1 e 2 k 2 η 2. Therefore, lim e(t) = lim η(t) =. t t 3.2 Improved Backstepping Design Intheclassic backsteppingdesign, thelyapunvovfunction is made decreasing at a constant rate. It may take a long time to have a desired tracking error if the initial error is relatively large. If we need a shorter time to get a desired error we have to choose larger constants k 1 and k 2, which 1148

3 17th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-11, 8 implies the increase of the control gains. How to achieve fast transient response without remarkably increasing the controller gains is abasic probleminpractice. Apossible solution is to exploit varying k 1 and k 2 which take larger values when the error is larger and take smaller values when the error is smaller, and asymptotically tend to some constants when time goes to infinity. This is our motivation to propose an improved backstepping design. We first recall the concept of class K functions. Definition (Krstic et al. (1995)). A function γ( ) : [, ) [, ) is called a class K function if it is continuous, strictly increasing and vanishes at the origin. Now, we give the improved backstepping design. Step 1. Let e = x 1 ω r. Choose constants k 1,k 2, and smooth class K functions α( ) and β( ) which are designfunctionsdeterminedbytherequiredresponse time and the maximum of control gains. Define the Lyapunov function for this step by V 1 = 1 2 e2, whose derivative along the system is u = Rx 2 + Mx 1 x 2 L [ 1 2Mx 2 J e + (k 2 + β(e))η +J d (k 1e + α(e)e)] = Rx 2 + Mx 1 x 2 L [ 1 2Mx 2 J e + (k 2 + β(e))η ( +(Mx 2 2 T L ) k 1 + α ) e e + α(e) ]. (4) Using the expression of η we can easily have the controller in terms of x 1,x 2 and e. Substituting the controller (4) in to (3) results in V 2 (k 1 + α(e))e 2 (k 2 + β(e))η 2. (5) Since k 1 >,k 2 >,α(e) and β(e), we have lim e(t) = lim η(t) =, t t which means that the objective of asymptotically speed tracking is fulfilled. Obviously, it holds that lim α(e(t)) = lim β(e(t)) =. t t V 1 = 1 J e(mx2 2 T L ). If we choose the virtual control x 2 2 as we will have Step 2. Set J M (k 1 + α(e))e + T L M, V 1 = (k 1 + α(e))e 2. η = M(x J M (k 1 + α(e))e T L M ) = Mx J(k 1 + α(e))e T L. Define the Lyapunov function for Step 2 as V 2 = V η2 = 1 2 e η2. Differentiating V 2 along the system yields ( M V 2 = e J x2 2 T ) L J + 2Mx 2η (u Rx 2 + Mx 1 x 2 ) L +ηj d (k 1e + α(e)e) = (k 1 + α(e))e 2 (k 2 + β(e))η 2 +η[ 1 J e + (k 2 + β(e))η + 2Mx 2 L (u (Rx 2 + Mx 1 x 2 )] +J d (k 1e + α(e)e)). Then, we design the controller (3) Therefore, lim (k 1 + α(e(t))) = k 1 t and lim (k 1 + β(e(t))) = k 2 t which means that the decay rate of V 2 tends to a constant. Remark 1. From (5) and in view of α and β being class K functions, we can see that V 2 decreases at a large rate when the tracking error is larger, which improves the transient response. On the other hand, smaller tracking error leads to smaller values of (k 1 + α) and (k 2 + β), which in turn reflects smaller controller gain. 4. JUMPING LOAD TORQUES When load changes, the corresponding load torque also changes. In this section, we will study the problem of speed tracking control for a series DC motor with jumping torques. Suppose that we have m different load torques in use. Then the the motor has m groups of dynamics: dx 1 dx 2 = M J x2 2 T j L J, j = 1,2,,m (6) = 1 L (Rx 2 + Mx 1 x 2 ) + 1 L u. For each j, (6) corresponds a torque and at each time t one and only one torque is connected to the motor. Suppose that at time t k one torque is switched off and another torque is switched on. Thus we have a switching sequence {t,t 1,, t k, }, which means that the i k -th torque is switched on at t k and switched off at t k+1. Therefore, we actually have a switched system: 1149

4 17th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-11, 8 dx 1 dx 2 = M J x2 2 T σ(t) L J, = 1 L (Rx 2 + Mx 1 x 2 ) + 1 L u. (7) where σ : [, ) {1,2, m} is a switching signal which is characterized by the switching sequence, i.e., σ(t) = i k when t [t k,t k+1 ). The system resulting from (7) when σ(t) is replaced by j is called the j-th subsystem of the switched system (7), which is exactly the j-th group dynamical equations of (6). Now we apply the improved backstepping design method in Section 3 to design a controller u j for the j-th subsystem of system (7) where u j = Rx 2 + Mx 1 x 2 L [ 1 2Mx 2 J e + (k 2 + β(e))η j ( +(Mx 2 2 T j L ) k 1 + α ) e e + α(e) ] η j = Mx J(k 1 + α(e))e T j L. Let V j 2 = 1 2 e η2 j. Then, when t [t k,t k+1 ), we have (8) V j 2 (k 1 + α(e))e 2 (k 2 + β(e))η 2 j (9) which guarantees asymptotic stability of each subsystem of the switched system (7). A switching controller is then constructed as u = u σ. Remark 2. In the discussion above, we construct different Lyapunov functions V j 2,j = 1,2,,m for individual subsystem. Moreover,for different load torques we maychoose different K functions α and β and different constants k 1 and k 2 to cope with different dynamical behavior. In this case, for the switched system (7), we have a switched Lyapunov function V σ(t) 2 = 1 2 e η2 σ(t), whose derivative along the system is given by V σ(t) 2 (k 1σ(t) + α σ(t) (e))e 2 (k 2σ(t) + β σ(t) (e))η 2 σ(t). (1) Although V σ(t) σ(t) 2 and V 2 are all given in terms of e and η σ(t), they can be easily expressed in x 1,x 2 and e. Remark 3. For jumping load torques, asymptotically speed tracking is impossible no matter what kind of controllers are applied. This can be seen from the dynamical equations of the motor. However, we can easily achieve acceptable tracking accurateness by properly designing feedback controllers for subsystems. This can be guaranteed by using multiple Lyapunov function principle for switched systems (Branicky (1998)). In the case that the switching law can be designed, we have more freedom to choose controllers for subsystems. 5. SIMULATIONS To demonstrate the effectiveness of the proposed design method we have conducted two groups of simulations for the cases of non-jumping load torque and jumping load torques, respectively. In each group, we use speed and currentresponses to compare the proposed improvedbeckstepping method with the classic backstepping method. Group 1: Non-jumping torque. The parameters are taken from Liu et al. (1999): R = 1Ω, J =.5kgm 2, L =.5H, M =.27H, T L = 55Nm, ω r = Figure 2 and Figure 3 show the speed and current responses respectively by the classic backstepping method with k 1 = k 2 = 1. Figure 4 and Figure 5 show the the speed and current responses respectively by the improved classic backstepping method with and k 1 = k 2 = 1 α(e) = β(e) =.5e 2. Group 2: Jumping load torques. Parameters: R = 1Ω, J =.5kgm 2, L =.5H, M =.27H, TL 1 = 55Nm, TL 2 = 11Nm ω r = The switching sequence is {,.5,1,1.5, } with the torque T L = 55Nm applied first. Figure 6 and Figure 7 show the speed and current responses respectively by the classic backstepping method with k 1 = k 2 = 1. Figure 8 and Figure 9 show the speed and current responses respectively by the improved classic backstepping method with k 1 = k 2 = 1 and α(e) = β(e) =.5e 2. These simulation results show that in any cases, the transient responses are substantially improved with the proposed control method. 115

5 17th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-11, Fig. 4. Improved backstepping Fig. 2. Classic backstepping Fig. 5. Improved backstepping Fig. 3. Classic backstepping CONCLUDING REMARKS We have proposed an improved backstepping speed controller for series DC motors. By adopting two class K functions as design functions, we achieved varying decay rates which gives fast transient response without remarkablyincreasingthecontrollergains. The proposed method can be obviously extended to the case that torque and the speed of the motor are unmeasurable by using adaptive mechanism. Jumping torques are also discussed where the improved backstepping method is more suitable. By expressing the dynamics of a series DC motor with jumping load torques as a switched system, we designed a switching controller Fig. 6. Classic backstepping with jumping torques 1151

6 17th IFAC World Congress (IFAC'8) Seoul, Korea, July 6-11, Fig. 7. Classic backstepping with jumping torques Fig. 8. Improved backstepping with jumping torques Fig. 9. Improved backstepping with jumping torques It is worth mentioning that the transient response for each load torque is more important than that in nonjumpingcase because eachtorque is only connected tothe motor for a finite time interval before it is switched off. Goodresponse ofeachtorqueoftenproduces gooracking effect. REFERENCES M.S.Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automatic Control, 43: , O.S.Bogosyan, M.Gokasan. A robust feedback linearizing approachforthe precise controlof direct drive systems. Proc. of IEEE International Conf. on Control Applications, pages: ,. M.J.Burridge, Z.Qu. An improved nonlinear control design for series DC motors. Computers and Electrical Engineering, 29: , 3. H.Chen, H.-B.Ji, B.Wang and H.-S.Xi. Coordinated passivation techniques for the dual-excited and steam-valving control of sunchronous generators. IEE Proceedings on Control Theory and Applications, 153:69-73, 6. J.Chiasson. Nonlinear differential geometric techniques for control of series DC motors. IEEE Trans. Control Systems Technology, 2:35-42, V.M.Hernandez, G.Herrera, B.Zuniga. Voltage regulation in series resonant DC-to-DC power converters with unknown resistive load. Proc. American Control Conference, pages: , 5. M.Krstic, I.Kanellakopoulos and P.Kokotovic. Nonlinear and Adaptive Control Design. John Wiley & Sons, Q.Liu. Robust tracking control of mechanical servo systems with inertial parameters varying in a large range. Proc. 1ST IEEE Conference on Industrial Electronics and Applications, pages: 1-5, 6. Z.Liu F.Luo and M.H.Rashid. Nonlinear speed controllers for series DC motor. Proc. IEEE Int.Conf. on Power Electronics and Drive Systems, pages: , 1999 Z. Li; C. Xia. Speed control of brushless DC motor based on CMAC and PID controller. Proc. of the Sixth World Congresson IntelligentControland Automation, pages: , 6. M.H.Rashid. Power Electronics: Circuits, Devices and Applications. Prentice-Hall, H.Sira-Ramirez. Design of PI controllers for DC-to-DC power supplies via extended linearization. International of Control, 51:1-6, 199. P.D.Olivier. Feedback linearization of DC motors. IEEE Trans.Industrail Electron, 38:498-51, R.-J.Wai. Robust control for induction servo motor drive. IEE Proceedingson Electr. Power Applications, 148: , 1. H.Wang, Y. Liu. Integrated design of speed-sensorless and adaptive speed controller for a brushless DC motor. IEEE Trans. on Power Electronics, 21: , 6. J.Yu, J.Hu, Jie Chang. The analysis of gain selection in adaptive backstepping motion control. proc. of 4th IEEE International Conference on Power Electronics and Drive Systems, pages: , 1. M. Zribl, A. AI-Zamel. Field weakening nonlinear control of a separately excited DC motor Control. Proc. of the FirstInternationalSymposiumon Communications and Signal Processing, pages: ,